Question

Formatted question description: https://leetcode.ca/all/1649.html

 1649. Create Sorted Array through Instructions

 Given an integer array instructions, you are asked to create a sorted array from the elements in instructions.
 You start with an empty container nums.
 For each element from left to right in instructions, insert it into nums.
 The cost of each insertion is the minimum of the following:

     The number of elements currently in nums that are strictly less than instructions[i].
     The number of elements currently in nums that are strictly greater than instructions[i].
     For example, if inserting element 3 into nums = [1,2,3,5], the cost of insertion is min(2, 1) (elements 1 and 2 are less than 3, element 5 is greater than 3) and nums will become [1,2,3,3,5].

 Return the total cost to insert all elements from instructions into nums.
 Since the answer may be large, return it modulo 109 + 7



 Example 1:

 Input: instructions = [1,5,6,2]
 Output: 1
 Explanation: Begin with nums = [].
 Insert 1 with cost min(0, 0) = 0, now nums = [1].
 Insert 5 with cost min(1, 0) = 0, now nums = [1,5].
 Insert 6 with cost min(2, 0) = 0, now nums = [1,5,6].
 Insert 2 with cost min(1, 2) = 1, now nums = [1,2,5,6].
 The total cost is 0 + 0 + 0 + 1 = 1.


 Example 2:

 Input: instructions = [1,2,3,6,5,4]
 Output: 3
 Explanation: Begin with nums = [].
 Insert 1 with cost min(0, 0) = 0, now nums = [1].
 Insert 2 with cost min(1, 0) = 0, now nums = [1,2].
 Insert 3 with cost min(2, 0) = 0, now nums = [1,2,3].
 Insert 6 with cost min(3, 0) = 0, now nums = [1,2,3,6].
 Insert 5 with cost min(3, 1) = 1, now nums = [1,2,3,5,6].
 Insert 4 with cost min(3, 2) = 2, now nums = [1,2,3,4,5,6].
 The total cost is 0 + 0 + 0 + 0 + 1 + 2 = 3.


 Example 3:

 Input: instructions = [1,3,3,3,2,4,2,1,2]
 Output: 4
 Explanation: Begin with nums = [].
 Insert 1 with cost min(0, 0) = 0, now nums = [1].
 Insert 3 with cost min(1, 0) = 0, now nums = [1,3].
 Insert 3 with cost min(1, 0) = 0, now nums = [1,3,3].
 Insert 3 with cost min(1, 0) = 0, now nums = [1,3,3,3].
 Insert 2 with cost min(1, 3) = 1, now nums = [1,2,3,3,3].
 Insert 4 with cost min(5, 0) = 0, now nums = [1,2,3,3,3,4].
 ​​​​​​​Insert 2 with cost min(1, 4) = 1, now nums = [1,2,2,3,3,3,4].
 ​​​​​​​Insert 1 with cost min(0, 6) = 0, now nums = [1,1,2,2,3,3,3,4].
 ​​​​​​​Insert 2 with cost min(2, 4) = 2, now nums = [1,1,2,2,2,3,3,3,4].
 The total cost is 0 + 0 + 0 + 0 + 1 + 0 + 1 + 0 + 2 = 4.


 Constraints:

     1 <= instructions.length <= 105
     1 <= instructions[i] <= 105

Algorithm

Using a tree array, single-point modification + interval query, tree[i] represents the number of occurrences of the number i in the inserted element, and the complexity of querying the number of less than/greater than x and inserting new elements during each traversal is 𝑂(𝑙𝑜𝑔𝑁), the total complexity is 𝑂(𝑁𝑙𝑜𝑔𝑁).

Code

Java

public class Create_Sorted_Array_through_Instructions {

    class Solution {
        public int createSortedArray(int[] instructions) {
            int cost = 0;
            int[] tree = new int[100001]; // tree[i]: num i count in after-insertion-array
            for (int i = 0; i < instructions.length; i++) {
                int num = instructions[i];
                int leftLen = sum(tree, num - 1);
                int rightLen = i - sum(tree, num); // index is i, so (i+1)-th then -1 to self exclude
                cost = (cost + Math.min(leftLen, rightLen)) % 1000000007;
                add(tree, num, 1, 100000);
            }
            return cost;
        }

        private int lowbit(int x) {
            return x & -x;
        }

        private void add(int[] tree, int index, int value, int max) {
            while (index <= max) {
                tree[index] += value;
                index += lowbit(index);
            }
        }

        private int sum(int[] tree, int index) {
            int sum = 0;
            while (index > 0) {
                sum += tree[index];
                index -= lowbit(index);
            }
            return sum;
        }
    }
}

Java

class Solution {
    final static int MODULO = 1000000007;
    final static int MAX = 100000;
    int[] segmentTree = new int[MAX * 2 + 1];

    public int createSortedArray(int[] instructions) {
        int cost = 0;
        int length = instructions.length;
        for (int i = 0; i < length; i++) {
            int num = instructions[i];
            int count = segmentTree[num + MAX] + 1;
            update(num, count);
            int sumRange1 = sumRange(1, num - 1), sumRange2 = sumRange(num + 1, MAX);
            cost = (cost + Math.min(sumRange1, sumRange2)) % MODULO;
        }
        return cost;
    }
    
    public void update(int i, int val) {
        int index = i + MAX;
        int prevVal = segmentTree[index];
        int difference = val - prevVal;
        while (index > 0) {
            segmentTree[index] = (segmentTree[index] + difference) % MODULO;
            index >>= 1;
        }
    }
    
    public int sumRange(int i, int j) {
        int sum = 0;
        int index1 = i + MAX, index2 = j + MAX;
        while (index1 <= index2) {
            if (index1 % 2 == 1) {
                sum = (sum + segmentTree[index1]) % MODULO;
                index1++;
            }
            if (index2 % 2 == 0) {
                sum = (sum + segmentTree[index2]) % MODULO;
                index2--;
            }
            index1 >>= 1;
            index2 >>= 1;
        }
        return sum;
    }
}

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